Prescribing coefficients of invariant irreducible polynomials

Abstract

Let F q be the finite field of q elements. We define an action of PGL(2, q) on F q [X] and study the distribution of the irreducible polynomials that remain invariant under this action for lower-triangular matrices. As a result, we describe the possible values of the coefficients of such polynomials and prove that, with a small finite number of possible exceptions, there exist polynomials of given degree with prescribed high-degree coefficients.

A group action on multivariate polynomials over finite fields

“…This was first noticed by Garefalakis in [5]. Recently, this action (and others related) has attracted attention from several authors (see [6], [4] and [2]), and some fundamental questions have been discussed such as the characterization and number of invariant irreducible polynomials of a given degree. The map induced by A preserves the degree of elements in I n (for n ≥ 2), but not in the whole ring F q [x]: for instance,…”